| $2.9$ Exercises By Theorem $2.7 .9$ it follows that for $a<b$, there exists a rational number between $a$ and $b$. Show there exists an integer $k$ such that $a<\frac{k}{2^{m}}<b$ for some $k, m$ integers. Show there is no smallest number in $(0,1)$. Recall $(0,1)$ means the real numbers which are strictly larger than 0 and smaller than 1 . Show there is no smallest number in $\mathbb{Q} \cap(0,1)$. Show that if $S \subseteq \mathbb{R}$ and $S$ is well ordered with respect to the usual order on $\mathbb{R}$ then $S$ cannot be dense in $\mathbb{R}$. Prove by induction that $\sum_{k-1}^{n} k^{3}=\frac{1}{4} n^{4}+\frac{1}{2} n^{3}+\frac{1}{4} n^{2}$. It is a fine thing to be able to prove a theorem by induction but it is even better to be able to come up with a theorem to prove in the first place. Derive a formula for $\sum_{k-1}^{n} k^{4}$ in the following way. Look for a formula in the form $A n^{5}+B n^{4}+$ $C n^{3}+D n^{2}+E n+F$. Then try to find the constants $A, B, C, D, E$, and $F$ such that things work out right. In doing this, show $$ \begin{gathered} (n+1)^{4}= \ \left(A(n+1)^{5}+B(n+1)^{4}+C(n+1)^{3}+D(n+1)^{2}+E(n+1)+F\right) \ -A n^{5}+B n^{4}+C n^{3}+D n^{2}+E n+F \end{gathered} $$ and so some progress can be made by matching the coefficients. When you get your answer, prove it is valid by induction. Prove by induction that whenever $n \geq 2, \sum_{k-1}^{n} \frac{1}{\sqrt{k}}>\sqrt{n}$. If $r \neq 0$, show by induction that $\sum_{k-1}^{n} a r^{k}=a \frac{r^{n+1}}{r-1}-a \frac{r}{r-1}$. Prove by induction that $\sum_{k-1}^{n} k=\frac{n(n+1)}{2}$. Let $a$ and $d$ be real numbers. Find a formula for $\sum_{k=1}^{n}(a+k d)$ and then prove your result by induction. Consider the geometric series, $\sum_{k-1}^{n} a r^{k-1}$. Prove by induction that if $r \neq 1$, then $\sum_{k-1}^{n} a r^{k-1}=\frac{a-a r^{n}}{1-r} .$ |
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